This is a question that I have been asked by both Seventh-Day Adventists (because they believe in the annihilation of the wicked) and atheists (who don't appear to stop and think that - in their worldview - all their loved ones have simply ceased to exist; is there no sorrow over that?). I expect that Jehovah's Witnesses ask it, too.
The answer to their question is, "No, Christians will not sorrow, whether for this or for any other reason." The Prophet Isaiah prophesied, "The ransomed of the LORD shall return and come to Zion with singing;
everlasting joy shall be upon their heads; they shall obtain gladness
and joy, and sorrow and sighing shall flee away" (Isaiah 35:10). And the Apostle John agreed (Revelation 21:4): "[God] will wipe away every tear from their eyes, and death shall be no
more, neither shall there be mourning, nor crying, nor pain anymore, for
the former things have passed away." These are general considerations; there shall be no sorrow, over any matter, in glory.
But specifically on this issue, there will be no sorrow. It is true that, in this life, we sorrow over such things, because we do not yet have the perspective we will have in our glorified state. We still view things according to the values of our fallen state, that is, we look at the situation as sinners. Yet, even now, the scriptures urge us to place the holiness and justice of God ahead of our distorted emotions. In Psalm 9:16, David advises, "The Lord has made Himself known; He has executed judgment; the wicked are snared in the work of their own hands." And in Psalm 51:4, the same writer tells of himself, "Against You, You only, have I sinned and done what is evil in Your sight, so that You may be justified in Your words and blameless in Your judgment." And further, in Psalm 58:10, David realizes, "The righteous will rejoice when he sees the vengeance; he will bathe his feet in the blood of the wicked." Notice that: when we are no longer sinners, in our glorification, we will have the perspective of God, and place His holiness ahead of our creaturely preferences. Logically, who matters more: God? Or your relatives?
David's point, in these Psalms, is that the justice of God's judgment is based on against whom sin is committed. Sin isn't naughtiness, as has become the common view in our society. It is an act of treason against our Creator, He Who made us and has provided the world we live in, the food, air, and water we require for survival, and the human comforts that make our lives enjoyable. To sin against Him after such gifts is wicked enough. However, if you further consider the gift that He has given in His Son, who suffered, bled, and died, how horrific we must now see sin to be. For this, God has said (Psalm 81:11-12), "My people did not listen to My voice; Israel would not submit to Me. So I gave them over to their stubborn hearts, to follow their own counsels." He has allowed us to pursue the consequences of our own choices and actions. Can there be any injustice in Him now?
The Puritan theologian Thomas Watson said, "The reason why sin committed in a short time is eternally punished is because every sin is committed against an infinite essence, and no less than eternity of punishment can satisfy. Why is treason punished with confiscation and death, but because it is against the king's person, which is sacred; much more that offense which is committed against God's crown and dignity is of a heinous and infinite nature, and cannot be satisfied with less than eternal punishment."
Yet, we must go further: God has allowed us, not only to pursue our own wicked choices, but also opportunities to awaken and repent of those choices. In Jeremiah 26:3 and 13, He said to Israel, "It may be they will listen, and every one turn from his evil way, that I
may relent of the disaster that I intend to do to them because of their
evil deeds... Now therefore mend your ways and your deeds, and obey the voice of the Lord your God, and the Lord will relent of the disaster that He has pronounced against you." And in Revelation 2:21, "I gave her [i. e., Jezebel] time to repent, but she refuses to repent of her sexual immorality." Has He been unfair? Unjust? Clearly, He hasn't. So, how can anyone claim to have been wronged when He finally carries out the judgment which He has, so far, withheld?
Now, to refer back to the question with which I began: Will Christians sorrow over loved-ones in Hell? No, we won't. Rather, we will rejoice that the holiness of God has been vindicated. In contrast, unbelievers will indeed suffer sorrow in the life to come, not just for their suffering loved-ones, but because of the judgment for their own personal sins. In fact, you will sorrow, not for loved-ones in Hell, but because of antipathy toward loved-ones in Heaven! If such sorrowing is a horror, something to be avoided, then the solution is to repent of your unbelief. Then, not only will you be freed from your condemnation, but you will also have a message of hope for your loved-ones. Rather than picture them in Hell, suffering for their sins, picture them saved from condemnation, never to suffer sorrow again. As John also says (Revelation 14:13, see also Job 3:17), "Blessed are the dead who die in the Lord from now on. 'Blessed indeed,'
says the Spirit, 'that they may rest from their labors!'"